Degenerate Diffusion in Heterogeneous Media

When

Start: 10/02/2013 - 4:15pm End : 10/02/2013 - 5:15pm

Category

Colloquium

Speaker

Guillermo Reyes, University of California, Irvine

Abstract

In this talk, I will present some recent results on the long-time behavior of non-negative solutions to the Cauchy problem for the Porous Medium Equation in the presence of variable density vanishing at infinity. More precisely, we consider the initial value problem

(P) in , in ,

where we assume >, ≥ and is positive, smooth and has a power-like decay at infinity, as for some >. The data are assumed to be non-negative and such that < (finite-energy solutions).

For >, the behavior of solutions depends on whether << or >. In the first case, the asymptotic behavior is described in terms of a one-parameter family of source-type, self-similar solutions of a related singular problem (so called Barenblatt-type solutions). For >, however, solutions to (P) have a universal long-time behavior in separate variables, typical of initial-boundary problems on bounded domains.
If has an intermediate decay, with <<, solutions still enjoy the finite propagation property (as in the case of lower ). In this range a more precise description may be given at the diffusive scale in terms of the Barenblatt-type solutions of the related singular equation .